This paper deals with the problem of approximating the infinite-dimensional algebraic Riccati equation, considered as an abstract equation in the Hilbert space of Hilbert-Schmidt operators. Two kinds of approximating schemes are proposed. The first scheme exploits the already established approximability of the corresponding dynamical Riccati equation together with its time convergence toward the steady state. The second method considers a particular sequence of finite-dimensional linear equations whose solutions are proved to converge toward the exact steady-state solution of the original problem.1. Introduction. Both linear quadratic (LQ) optimal control and optimal linear filtering problems for linear systems evolving in Hilbert spaces lead to an infinitedimensional Riccati equation. This has motivated the wide interest that, for at least two decades, has been devoted to establishing conditions for the existence and uniqueness of the solution of this equation [6], [9], [10], [12]. This problem has also been considered in [8], [15], [16], [20], [26], and [32] with particular reference to the LQ optimal control, in [6], [7], [19], [29], and [36] with reference to the optimal linear filtering, and in [10] and [11] with reference to both cases. In the above papers the topic is treated in different settings according to the different forms that the infinitedimensional Riccati equation can take, depending on the structure assumed for the system dynamics. Particularly important is the so-called infinite-horizon problem, which arises when the Riccati equation admits a steady-state solution. The corresponding nondynamical equation is referred to as the algebraic Riccati equation (ARE) [11], [18], [20], and [38]. Unfortunately, in this case the approximation problem tends to be much more difficult than the finite-horizon problem, for which significant results are available. Such a problem was discussed in [3], [21], and [22] with reference to the LQ optimal control, under the hypothesis that both the approximate semigroups and their adjoints